|The short preprint
The super-Sasaki metric on the antitangent bundle, has now been extensivly rewriten and improved. I wrote about the first draft here.
One of the major improvments is that I now clearly link the construction to almost Hermitian manifolds, although the almost complex structure plays no direct role in the lifting the metric and almost sympletic structure to the antitangent bundle.
I now also give a concrete example of the super-Sasaki metric with Misner space as the underlying pseudo-Riemannian manifold. As Misner space is topologically the infinite cylinder it is canonically a symplectic manifold. To my knowedge, the Riemannian metric I construct on the antitangent bundle of Misner space is completley new.
I will briefly remark that Misner space has been used as a toy space-time to investigate various conjectures in general relativity and semi-classical gravity. This space has closed timelike curves, known as CTCs, or to the layman, time machines. It was studying quantum fields on this space that led Hawking to his chronological protection conjecture.
There are, of course, lots of open questions here. It would be interesting to see how much of the classical theory of Sasaki carries over to this super-setting.